Push-Sum With Transmission Failures

Abstract

The push-sum algorithm allows distributed computing of the average on a directed graph, and is particularly relevant when one is restricted to one-way and/or asynchronous communications. We investigate its behavior in the presence of unreliable communication channels where messages can be lost. We show that exponential convergence still holds and deduce fundamental properties that implicitly describe the distribution of the final value obtained. We analyze the error of the final common value we get for the essential case of two nodes, both theoretically and numerically. We provide performance comparison with a standard consensus algorithm. * B. Gerencsér and J. M. Hendrickx are with ICTEAM Institute, Université catholique de Louvain, Belgium balazs.gerencser@uclouvain.be and julien.hendrickx@uclouvain.

Tight Bounds on the Convergence Rate of Generalized Ratio Consensus Algorithms

“…The key results of this paper are stated as Theorems 12, 14, 16 and 19, extending previous results on the almost sure exponential convergence in the context of ratio consensus such as given in [9] and [10], in particular providing upper bounds for the almost sure exponential convergence rate in terms of spectral gaps associated with stationary sequences of matrices. It will be shown that these upper bounds are sharp in Theorem 21, thus solving an open problem formulated in the conclusion of one of the fundamental papers [2] under very general conditions, quoting from their Conclusion: "The next step of this work is to compute analytically the speed of convergence of Weighted Gossip.…”

“…In this section we summarize the implications of the above stated results for the classic push-sum or weighted gossisp algorithm, allowing packet loss as described in the Introduction, which is in line with the setting of [10].…”

“…The limit is random, in contrast to the case of classic push-sum or weighted gossip algorithms without packet loss. On the other hand, there is ample empirical evidence that decreasing the probability of packet loss leads to higher concentration of the distribution of π T x, aroundx, see [10].…”

“…of (35) be denoted by y n and z n , respectively. The elementary lemma below, which will be used later on, has been established in [10] for the case of the push-sum algorithm with packet loss: Lemma 18. The values y n and z n are monotone non-decreasing and non-increasing, respectively.…”

Fully Asynchronous Push-Sum With Growing Intercommunication Intervals

“…This algorithm has been widely used to develop protocols that reach average consensus, under different assumptions and scenarios; such as the presence of bounded delays [5], time varying graphs [6][7], or asynchronous communication [8].Since reliable communication is a very restrictive assumption in network applications, or expensive to enforce, recent work has considered algorithms that reach consensus in a setting where communication between agents is unreliable. While in this case, push-sum might not converge to average, exponential convergence still holds and the error between the final value and the true average can be characterized [9]. In [10], Vaidya et al introduce the technique of running sums (counters) and modify push-sum to overcome possible packet drops and imprecise knowledge of the network in a synchronous communication setting.…”